Compute the exact value of the expression $\left|\pi - | \pi - 7 | \right|$.  Write your answer using only integers and $\pi$, without any absolute value signs.
Answer: We begin by examining the quantity $|\pi - 7|$.  Since $\pi$ is less than 4, clearly $\pi-7$ will be negative.  Hence we must negate this quantity to obtain its absolute value, which is always positive.  In other words, \[ |\pi - 7| = -(\pi - 7) = 7- \pi. \]Continuing, we next consider the expression $\pi-|\pi - 7|$, which reduces to $2\pi - 7$ in light of the above computation.  Since $\pi$ is less than 3.5, this quantity is also negative.  Hence we must negate it just as before when taking absolute value, leading to our final answer of $\boxed{7-2\pi}.$